Question

Please help me prove the identity​

Answer 1

`(1-\tan x )/(1+\cot x)`

Explanation:

Let
`(\sin 2x+\cos 2x-1)/(\sin 2x+\cos 2x + 1)`, we proceed to prove the given identity by trigonometric and algebraic means:

1)
`(\sin 2x+\cos 2x-1)/(\sin 2x+\cos 2x + 1)` Given

2)
`(2\cdot \sin x \cdot \cos x +\cos ^(2)x - \sin^(2)x-1)/(2\cdot \sin x \cdot \cos x +\cos ^(2)x - \sin^(2)x+1)`
`\sin 2x = 2\cdot \sin x \cdot \cos x`/
`\cos 2x = \cos^(2)x - \sin^(2)x`

3)
`(2\cdot \sin x \cdot \cos x -\sin^(2)x-(1-\cos^(2)x))/(2\cdot \sin x\cdot \cos x +\cos^(2)x+(1-\sin^(2)x))` Commutative, associative and distributive properties/
`-a = (-1)\cdot a`

4)
`(2\cdot \sin x \cdot \cos x -2\cdot \sin^(2)x)/(2\cdot \sin x \cdot \cos x +2\cdot \cos^(2)x)`
`\sin^(2)x + \cos^(2)x = 1`

5)
`((2\cdot \sin x)\cdot (\cos x-\sin x))/((2\cdot \cos x)\cdot (\sin x +\cos x))` Distributive and associative properties.

6)
`(\sin x\cdot (\cos x-\sin x))/(\cos x\cdot (\sin x +\cos x))` Existence of multiplicative inverse/Commutative and modulative properties.

7)
`((\cos x -\sin x)/(\cos x) )/((\sin x + \cos x)/(\sin x) )`
`((x)/(y) )/((w)/(z) ) = (x\cdot z)/(y\cdot w)`

8)
`((\cos x)/(\cos x)-(\sin x)/(\cos x) )/((\sin x)/(\sin x)+(\cos x)/(\sin x) )`
`(x+y)/(w) = (x)/(w) + (y)/(w)`

9)
`(1-\tan x )/(1+\cot x)` Existence of multiplicative inverse/
`\tan x = (\sin x)/(\cos x)`/
`\cot x = (\cos x)/(\sin x)`/Result